GATE 2025 ST – Question 17

Stochastic Processes · Poisson and Birth-Death Processes · 1 mark · Multiple choice

N(t) is a Poisson process of rate 2. Let X=N(6)−N(1),Y=N(5)−N(3),W=N(6)−N(5),Z=N(3)−N(1). Which holds?

Cov(W,Z)=2
Y+Z~Poisson(10)
P(Y=Z)=1
Cov(X,Y)=4

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Correct answer: (D) Cov(X,Y)=4

Explanation

W,Y,Z count disjoint intervals and are independent with means 2,4,4. X=W+Y+Z, so Cov(X,Y)=Var(Y)=4. Also Cov(W,Z)=0 and Y+Z~Poisson(8). Independent nondegenerate Y,Z are not always equal. **D**.

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